报警-趋向性模型的经典解与广义解
Classical and generalized solutions of an alarm-taxis model
AI总结：
本文研究二维有界区域中猎物、捕食者与超级捕食者的报警-趋向性模型，证明纯报警-趋向性模型存在全局经典解，并在完整模型中克服趋向项和耦合项带来的正则性困难，获得全局广义解。
摘要：
在有界、空间二维区域中，系统 \begin{equation*} \left\lbrace\begin{alignedat}{3} u_t &= d_1 Δu && &&+ u(λ_1 - μ_1 u - a_1 v - a_2 w), \\\\ v_t &= d_2 Δv &&- ξ\nabla \cdot (v \nabla u) &&+ v(λ_2 - μ_2 v + b_1 u - a_3 w),\\\\ w_t &= d_3 Δw &&- χ\nabla \cdot (w \nabla (uv)) &&+ w(λ_3 - μ_3 w + b_2 u + b_3 v), \end{alignedat}\right. \end{equation*} 配以初值和齐次 Neumann 边界条件，刻画了猎物（密度为 $u$）、捕食者（密度为 $v$）和超级捕食者（密度为 $w$）之间的相互作用，其中超级捕食者捕食另外两个种群。除了随机运动和初级捕食者的猎物趋向行为之外，该系统的关键特征在于次级捕食者会对猎物发出的报警信号作出反应，而猎物在受到初级捕食者攻击时发出这些信号。我们首先证明在纯报警-趋向性模型（即 $ξ= 0$）中，全局经典解存在。对于完整模型（$ξ> 0$），趋向项以及第一个方程中 $-a_2 uw$ 项的存在显然阻碍了某些 bootstrap 过程，这意味着可用的正则性信息相当有限。尽管如此，我们仍能获得全局广义解。一个重要的技术挑战是保证前两个解分量的（加权）梯度的强收敛，以便得出近似解收敛到极限问题的广义解的结论。
英文摘要：
In bounded, spatially two-dimensional domains, the system \begin{equation*} \left\lbrace\begin{alignedat}{3} u_t &= d_1 Δu && &&+ u(λ_1 - μ_1 u - a_1 v - a_2 w), \\ v_t &= d_2 Δv &&- ξ\nabla \cdot (v \nabla u) &&+ v(λ_2 - μ_2 v + b_1 u - a_3 w),\\ w_t &= d_3 Δw &&- χ\nabla \cdot (w \nabla (uv)) &&+ w(λ_3 - μ_3 w + b_2 u + b_3 v), \end{alignedat}\right. \end{equation*} complemented with initial and homogeneous Neumann boundary conditions, models the interaction between prey (with density $u$), predator (with density $v$) and superpredator (with density $w$), which preys on both other populations. Apart from random motion and prey-tactical behavior of the primary predator, the key aspect of this system is that the secondary predator reacts to alarm calls of the prey, issued by the latter whenever attacked by the primary predator. We first show in the pure alarm-taxis model, i.e. if $ξ= 0$, that global classical solutions exist. For the full model (with $ξ> 0$), the taxis terms and the presence of the term $-a_2 uw$ in the first equation apparently hinder certain bootstrap procedures, meaning that the available regularity information is rather limited. Nonetheless, we are able to obtain global generalized solutions. An important technical challenge is to guarantee strong convergence of (weighted) gradients of the first two solution components in order to conclude that approximate solutions converge to a generalized solution of the limit problem.
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